Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-6x+y &= 7 \\ -x-y &= 1\end{align*}$
Begin by moving the $x$ -term in the second equation to the right side of the equation. $-y = x+1$ Divide both sides by $-1$ to isolate $y$ $y = {-x - 1}$ Substitute this expression for $y$ in the first equation. $-6x+({-x - 1}) = 7$ $-6x - x - 1 = 7$ Simplify by combining terms, then solve for $x$ $-7x - 1 = 7$ $-7x = 8$ $x = -\dfrac{8}{7}$ Substitute $-\dfrac{8}{7}$ for $x$ back into the top equation. $-6( -\dfrac{8}{7})+y = 7$ $\dfrac{48}{7}+y = 7$ $y = \dfrac{1}{7}$ $y = \dfrac{1}{7}$ The solution is $\enspace x = -\dfrac{8}{7}, \enspace y = \dfrac{1}{7}$.